Lesson 5: The Graph of the Equation y = f(x)
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1 Lesson 5: The Graph of the Equation y = f(x) Learning targets: I can identify when a function is increasing, decreasing, positive and negative and use interval notation to describe intervals where the function is increasing or decreasing, positive and negative Opening Activity: Write the solution set to the following number lines in interval notation: a) b) Interval Notation: Interval Notation: Explaining the Symbols c) 3 < x < 5 d) x 3 Interval Notation: Interval Notation: e) x > 3 f) x < 4 or x 2 Interval Notation: Interval Notation: Example 1 Express each of the following using interval notation. Function Vocabulary: Increasing: A function is increasing if, as the x-values increase from left to right, the y-values increase. (Use interval notation to describe the x-values where this happens.) Decreasing: A function is decreasing if, as the x-values increase from left to right, the y-values decrease. (Use interval notation to describe the x-values where this happens.) Relative Maximum: A point where a function changes from increasing to decreasing (a peak) Relative Minimum: A point where a function changes from decreasing to increasing (a valley)
2 Example 2: At right is an example of a polynomial function. a) Which points (A, B, C, or D) are relative maximums? b) Which points (A, B, C, or D) are relative minimums? c) Name the interval(s) where the function is increasing. d) Name the interval(s) where the function is decreasing. Example 3: At right is the graph of a quadratic function. a.) What is the domain? b.) What is the range? c) At what interval is the function increasing? d) At what interval is the function decreasing? e) Determine whether the quadratic has a relative maximum or relative minimum. Function Vocabulary: Positive: A function is positive if the function is above the x-axis. Negative: A function is negative if the function is below the x-axis. **NOTE: zero is neither positive, nor negative so at the point where the function crosses the x-axis, a parenthesis should surround that number!! x-intercept(s): The coordinates where a function crosses the x-axis y-intercept: The coordinates where a function crosses the y-axis (there can only be 1)
3 Example 4: At right is an example of a polynomial function. a) What is the y-intercept? b) What is the x-intercept(s)? c) Name the interval(s) where the function is positive. d) Name the interval(s) where the function is negative. e) What is the domain and range for this graph? Example 5 : a. What is the y-intercept of the graph? b. What is the x-intercept of the graph? c. Over what intervals is the function positive? d. Over what intervals is the function negative?
4 Lesson 5: The Graph of the Equation y = f(x) Classwork Exercise 1. Express each of the following using interval notation. Exercise 2: Below is an example of a piece-wise function. a) Label any relative maximum with an A. b) Label any relative minimum with a B. c) Name the interval(s) where the function is increasing. d) Name the interval(s) where the function is decreasing. e) How do you know that this is a function? f) State the domain in set notation and interval notation. g) State the domain in set notation and interval notation
5 Lesson 5: The Graph of the Equation y = f(x)
6 2. Below is the graph of a quadratic function. Describe the intervals where the function is positive/negative. State the x and y intercepts of the function. Review: state the domain and range in interval notation. Intervals Positive: Negative: X Intercepts: Y Intercept: Domain: Range:
7 3: a. What is the y-intercept of the graph? b. What are the x-intercepts of the graph? c. Over what intervals is the function positive? d. Over what intervals is the function negative? Challenge - 4 : For the graph below, identify the increasing and decreasing intervals, the positive and negative intervals, and the intercepts. (Pay attention that there are no arrows at the end of the function).
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